Quantitative exponential mixing for the randomized Chirikov standard map

Yankai Shi; Ziyu Liu

arxiv: 2605.20702 · v1 · pith:GABILSKQnew · submitted 2026-05-20 · 🧮 math.PR · math.DS

Quantitative exponential mixing for the randomized Chirikov standard map

Ziyu Liu , Yankai Shi This is my paper

Pith reviewed 2026-05-21 02:51 UTC · model grok-4.3

classification 🧮 math.PR math.DS

keywords random dynamical systemsexponential mixingChirikov standard mapincompressible flowsquantitative ergodicityenhanced dissipationrandomized maps

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The pith

Randomized Chirikov standard map shows almost-sure quantitative exponential mixing for large kicking strengths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that randomizing the Chirikov standard map on the two-torus overcomes the obstructions to global ergodicity seen in the deterministic case, producing explicit almost-sure quantitative exponential mixing when kicking strengths are sufficiently large. A sympathetic reader would care because this supplies concrete rates at which the system loses memory of its starting point, useful for predicting long-term statistics in chaotic systems. The authors create a criterion for incompressible random dynamical systems that reduces proving quantitative exponential mixing to checking several verifiable conditions. They also obtain qualitative exponential mixing and enhanced dissipation from a milder condition on the kicking strength.

Core claim

We establish explicit almost-sure quantitative exponential mixing for the randomized Chirikov standard map on the two-torus when kicking strengths are sufficiently large. This is achieved by formulating a criterion for incompressible random dynamical systems that reduces quantitative exponential mixing to several verifiable conditions. We additionally derive qualitative exponential mixing and enhanced dissipation under a milder parameter condition.

What carries the argument

Criterion for incompressible random dynamical systems that reduces quantitative exponential mixing to verifiable conditions on the map.

If this is right

The randomized map exhibits almost-sure quantitative exponential mixing with explicit rates when kicking strengths are large enough.
The mixing holds for almost every realization of the randomness.
Qualitative exponential mixing and enhanced dissipation follow from a milder condition on the kicking strength.
The criterion provides a general reduction applicable to this randomized map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The criterion could be applied to other randomized incompressible maps to obtain quantitative mixing without separate case analysis.
Explicit rates would allow estimates of relaxation times when the map is used in statistical sampling or numerical experiments.
The result indicates that adding randomness can bypass invariant structures that block mixing in the deterministic Chirikov map.
The link between quantitative mixing and enhanced dissipation suggests possible extensions to transport or dissipation problems in related random systems.

Load-bearing premise

Kicking strengths are sufficiently large, since both the quantitative rates and the reduction to verifiable conditions depend on this regime.

What would settle it

A calculation or simulation for some sufficiently large kicking strength that shows the correlation functions do not decay exponentially at the claimed rate, or that one of the verifiable conditions fails.

read the original abstract

We investigate the mixing properties of a randomized Chirikov standard map on $\mathbb{T}^2$. While the deterministic dynamics exhibit obstructions to global ergodicity, we establish explicit almost-sure quantitative exponential mixing when kicking strengths are sufficiently large. To achieve this, we formulate a criterion for incompressible random dynamical systems, reducing quantitative exponential mixing to serval verifiable conditions. Additionally, we provide a milder parameter condition to derive qualitative exponential mixing and enhanced dissipation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable criterion that turns quantitative mixing for incompressible random dynamical systems into checkable conditions and applies it to get almost-sure exponential rates for the randomized Chirikov map under large kicks.

read the letter

The central new piece is the criterion for incompressible random dynamical systems. It reduces the job of proving quantitative exponential mixing to a short list of verifiable conditions on expansion, minorization, and tail control. They then carry this out for the randomized Chirikov map and obtain explicit almost-sure rates once the kick strength exceeds some threshold. A milder version of the same conditions yields qualitative mixing plus enhanced dissipation. That separation is useful and keeps the argument organized rather than ad hoc. The deterministic obstructions to ergodicity are sidestepped cleanly by the randomization, which is the right move here. The application to this particular map is concrete and gives rates instead of just existence, which is an improvement over many earlier qualitative results in random dynamics. The soft spot is the parameter regime. The abstract and the criterion both lean on “sufficiently large” kicks, but the letter does not make the threshold explicit or show how the constants behave as the strength grows. For the almost-sure claim to deliver a uniform rate, the lower bounds on Lyapunov exponents and the Doeblin constants must not deteriorate on sets of positive probability. The stress-test concern about uniformity is reasonable; if the verification only works pathwise with constants that depend on the realization, the explicit rate may not survive taking the almost-sure limit. The paper appears to control tails, yet the dependence on the kick distribution still needs to be tracked more carefully to close the argument. This work is aimed at people who already care about quantitative rates in random dynamical systems and ergodic theory. A reader looking for a reusable reduction or for explicit bounds on a standard example will get something out of it. It is worth sending to a serious referee. The criterion looks portable enough that the details should be checked carefully rather than desk-rejected.

Referee Report

2 major / 2 minor

Summary. The paper investigates mixing for a randomized Chirikov standard map on the 2-torus. It claims to prove explicit almost-sure quantitative exponential mixing when the kicking strength is sufficiently large, by introducing a general criterion for incompressible random dynamical systems that reduces the mixing property to several verifiable conditions on the RDS. A milder parameter regime is also treated to obtain qualitative exponential mixing together with enhanced dissipation.

Significance. If the central claims hold with the stated explicit rates, the work would supply quantitative control on mixing rates for a stochastic perturbation of a map known to have ergodicity obstructions in the deterministic case. The reduction of quantitative mixing to verifiable RDS conditions could serve as a reusable tool in random dynamical systems. The almost-sure nature of the rates and the explicit dependence on kicking strength are potentially strong features, provided uniformity over realizations is established.

major comments (2)

[§3] §3 (criterion for incompressible RDS): the minorization and expansion conditions are stated to be verifiable, yet the transfer to an almost-sure quantitative rate for the Chirikov map requires a uniform lower bound on the Lyapunov exponent that holds with probability 1; the manuscript does not exhibit an explicit tail estimate on the kick distribution that guarantees this uniformity independently of the realization.
[Theorem 1.1] Theorem 1.1 (main quantitative statement): the claimed exponential rate is asserted to be almost sure and explicit once the kicking strength exceeds a threshold, but the proof reduces the rate to constants whose dependence on the random kicks is not shown to remain bounded almost surely; without this control the explicit rate may deteriorate on a null set of positive measure.

minor comments (2)

[Abstract] Abstract, line 3: 'serval' is a typographical error for 'several'.
[§2] Notation for the random kick distribution and the associated probability space should be introduced once at the beginning of §2 and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to the major comments point by point below, indicating where we agree and what revisions we will make to strengthen the presentation of the almost-sure quantitative results.

read point-by-point responses

Referee: [§3] §3 (criterion for incompressible RDS): the minorization and expansion conditions are stated to be verifiable, yet the transfer to an almost-sure quantitative rate for the Chirikov map requires a uniform lower bound on the Lyapunov exponent that holds with probability 1; the manuscript does not exhibit an explicit tail estimate on the kick distribution that guarantees this uniformity independently of the realization.

Authors: We agree that an explicit tail estimate on the kick distribution would make the uniformity of the Lyapunov exponent lower bound fully rigorous and independent of realizations. In the current manuscript, the criterion assumes such a bound holds with probability 1, and we verify the conditions for the Chirikov map under large kicking strengths. To address this, we will add in the revision a new lemma (e.g., in an appendix) that provides an explicit tail probability estimate ensuring that the Lyapunov exponent is bounded below by a positive constant (depending only on the kicking strength threshold) with probability 1. This estimate will rely on the moment assumptions on the random kicks and standard concentration inequalities. revision: yes
Referee: [Theorem 1.1] Theorem 1.1 (main quantitative statement): the claimed exponential rate is asserted to be almost sure and explicit once the kicking strength exceeds a threshold, but the proof reduces the rate to constants whose dependence on the random kicks is not shown to remain bounded almost surely; without this control the explicit rate may deteriorate on a null set of positive measure.

Authors: The referee correctly identifies that the explicit rate in Theorem 1.1 is expressed in terms of constants that depend on the random kicks. However, the general criterion introduced in §3 is constructed so that these constants remain bounded almost surely when the verifiable conditions (including the uniform Lyapunov exponent lower bound) are satisfied. In the proof of Theorem 1.1, we show that for sufficiently large kicking strength, these conditions hold almost surely. To make this transparent, we will revise the proof to include a step explicitly bounding the constants uniformly using the tail estimates mentioned above. We do not believe the rate deteriorates on a positive measure set, as the null set is controlled by the probability 1 event where the conditions hold. revision: partial

Circularity Check

0 steps flagged

Derivation reduces to independent verifiable conditions with no self-referential reductions

full rationale

The abstract states that quantitative exponential mixing is reduced to several verifiable conditions on incompressible random dynamical systems. No equations or steps in the provided context show a prediction or result that equals its inputs by construction, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The criterion is presented as external and checkable rather than tautological. This is the most common honest non-finding for papers that explicitly separate their main claim from fitted or self-defined quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background results in random dynamical systems and incompressible flows; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard properties of incompressible random dynamical systems on the torus.
Invoked to formulate the mixing criterion.

pith-pipeline@v0.9.0 · 5584 in / 1085 out tokens · 35808 ms · 2026-05-21T02:51:44.616285+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1 ... uniform contraction estimates (3.1) ... quantitative small set condition (3.3) ... quantitative Harris theorem
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lyapunov function V(x,y):=dist(x,y)^{-p} ... drift condition P^{(2),m}V ≤ γV + C
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

topological irreducibility ... approximate controllability ... four-step Jacobian full rank at z^*

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

[1]

Alberti, G

G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing by incompressible flows.J. Amer. Math. Soc., 32(2):445–490, 2019

work page 2019
[2]

L. Arnold. Random dynamical systems. InDynamical systems (Montecatini Terme, 1994), volume 1609 of Lecture Notes in Math., pages 1–43. Springer, Berlin, 1995

work page 1994
[3]

P. H. Baxendale and D. W. Stroock. Large deviations and stochastic flows of diffeomorphisms.Probab. Theory Related Fields, 80(2):169–215, 1988

work page 1988
[4]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Almost-sure enhanced dissipation and uniform-in- diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes.Probab. Theory Related Fields, 179(3-4):777–834, 2021

work page 2021
[5]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations.Ann. Probab., 50(1):241–303, 2022

work page 2022
[6]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Lagrangian chaos and scalar advection in stochastic fluid mechanics.J. Eur. Math. Soc. (JEMS), 24(6):1893–1990, 2022

work page 1990
[7]

Blumenthal, M

A. Blumenthal, M. Coti Zelati, and R. S. Gvalani. Exponential mixing for random dynamical systems and an example of Pierrehumbert.Ann. Probab., 51(4):1559–1601, 2023

work page 2023
[8]

A. Bressan. A lemma and a conjecture on the cost of rearrangements.Rend. Sem. Mat. Univ. Padova, 110:97–102, 2003

work page 2003
[9]

B. V. Chirikov.Research concerning the theory of non-linear resonance and stochasticity. CERN, Geneva,

work page
[10]

Translated at CERN from the Russian (IYAF-267-TRANS-E)

work page
[11]

B. V. Chirikov. A universal instability of many-dimensional oscillator systems.Phys. Rep., 52(5):264–379, 1979

work page 1979
[12]

M. Christ. Hilbert transforms along curves. I. Nilpotent groups.Ann. of Math. (2), 122(3):575–596, 1985

work page 1985
[13]

Constantin, A

P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoˇ s. Diffusion and mixing in fluid flow.Ann. of Math. (2), 168(2):643–674, 2008

work page 2008
[14]

Cooperman, G

W. Cooperman, G. Iyer, and S. Son. A Harris theorem for enhanced dissipation, and an example of Pierre- humbert.Nonlinearity, 38(4):Paper No. 045027, 32, 2025

work page 2025
[15]

Cooperman and K

W. Cooperman and K. Rowan. Exponential scalar mixing for the 2D Navier-Stokes equations with degenerate stochastic forcing.Invent. Math., 243(3):863–959, 2026

work page 2026
[16]

Coti Zelati, T

M. Coti Zelati, T. M. Elgindi, and K. Widmayer. Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow.Comm. Math. Phys., 378(2):987–1010, 2020

work page 2020
[17]

Coti Zelati and V

M. Coti Zelati and V. Navarro-Fern´ andez. Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows.J. Dynam. Differential Equations, 2026, early access

work page 2026
[18]

Crippa and C

G. Crippa and C. De Lellis. Estimates and regularity results for the DiPerna-Lions flow.J. Reine Angew. Math., 616:15–46, 2008

work page 2008
[19]

Da Prato and J

G. Da Prato and J. Zabczyk.Ergodicity for infinite-dimensional systems. Cambridge University Press, Cam- bridge, 1996

work page 1996
[20]

Dolgopyat, V

D. Dolgopyat, V. Kaloshin, and L. Koralov. Sample path properties of the stochastic flows.Ann. Probab., 32(1A):1–27, 2004

work page 2004
[21]

P. Duarte. Plenty of elliptic islands for the standard family of area preserving maps.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 11(4):359–409, 1994

work page 1994
[22]

T. M. Elgindi and A. Zlatoˇ s. Universal mixers in all dimensions.Adv. Math., 356:106807, 33, 2019

work page 2019
[23]

Furstenberg

H. Furstenberg. Noncommuting random products.Trans. Amer. Math. Soc., 108:377–428, 1963

work page 1963
[24]

T. Gallay. Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices.Arch. Ration. Mech. Anal., 230(3):939–975, 2018

work page 2018
[25]

Gorodetski

A. Gorodetski. On stochastic sea of the standard map.Comm. Math. Phys., 309(1):155–192, 2012

work page 2012
[26]

J. M. Greene. A method for determining a stochastic transition.Journal of Mathematical Physics, 20(6):1183– 1201, 1979

work page 1979
[27]

Hairer and J

M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing.Ann. of Math. (2), 164(3):993–1032, 2006

work page 2006
[28]

Hairer and J

M. Hairer and J. C. Mattingly. Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations.Ann. Probab., 36(6):2050–2091, 2008

work page 2050
[29]

Hairer and J

M. Hairer and J. C. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs.Electron. J. Probab., 16:no. 23, 658–738, 2011

work page 2011
[30]

Hairer and J

M. Hairer and J. C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. InSeminar on Stochastic Analysis, Random Fields and Applications VI, volume 63 ofProgr. Probab., pages 109–117. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011
[31]

G. Iyer, A. Kiselev, and X. Xu. Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows.Nonlinearity, 27(5):973–985, 2014. MIXING FOR THE CHIRIKOV MAP 37

work page 2014
[32]

Iyer and H

G. Iyer and H. Zhou. Quantifying the dissipation enhancement of cellular flows.SIAM J. Math. Anal., 55(6):6496–6516, 2023

work page 2023
[33]

Jurdjevic.Geometric control theory, volume 52 ofCambridge Studies in Advanced Mathematics

V. Jurdjevic.Geometric control theory, volume 52 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997

work page 1997
[34]

Kiselev, R

A. Kiselev, R. Shterenberg, and A. Zlatoˇ s. Relaxation enhancement by time-periodic flows.Indiana Univ. Math. J., 57(5):2137–2152, 2008

work page 2008
[35]

Ledrappier

F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. InLyapunov exponents (Bre- men, 1984), volume 1186 ofLecture Notes in Math., pages 56–73. Springer, Berlin, 1986

work page 1984
[36]

Lin, J.-L

Z. Lin, J.-L. Thiffeault, and C. R. Doering. Optimal stirring strategies for passive scalar mixing.J. Fluid Mech., 675:465–476, 2011

work page 2011
[37]

R. S. MacKay. A renormalisation approach to invariant circles in area-preserving maps. volume 7, pages 283–300. 1983. Order in chaos (Los Alamos, N.M., 1982)

work page 1983
[38]

R. S. MacKay, J. D. Meiss, and I. C. Percival. Transport in Hamiltonian systems.Phys. D, 13(1-2):55–81, 1984

work page 1984
[39]

Mathew, I

G. Mathew, I. Mezi´ c, and L. Petzold. A multiscale measure for mixing.Phys. D, 211(1-2):23–46, 2005

work page 2005
[40]

Meyn and R

S. Meyn and R. L. Tweedie.Markov chains and stochastic stability. Cambridge University Press, Cambridge, second edition, 2009. With a prologue by Peter W. Glynn

work page 2009
[41]

Navarro-Fern´ andez and C

V. Navarro-Fern´ andez and C. Seis. Exponential mixing by random cellular flows.J. Funct. Anal., 290(2):Paper No. 111227, 2026

work page 2026
[42]

C. Seis. Maximal mixing by incompressible fluid flows.Nonlinearity, 26(12):3279–3289, 2013

work page 2013
[43]

S. Son. Quantitative dependence of the pierrehumbert flow’s mixing rate on the amplitude.arXiv:2410.27686, 2025

work page arXiv 2025
[44]

Thiffeault

J.-L. Thiffeault. Using multiscale norms to quantify mixing and transport.Nonlinearity, 25(2):R1–R44, 2012

work page 2012
[45]

D. Wei. Diffusion and mixing in fluid flow via the resolvent estimate.Sci. China Math., 64(3):507–518, 2021

work page 2021
[46]

D. Wei, Z. Zhang, and W. Zhao. Linear inviscid damping and enhanced dissipation for the Kolmogorov flow. Adv. Math., 362:106963, 103, 2020

work page 2020
[47]

Yao and A

Y. Yao and A. Zlatoˇ s. Mixing and un-mixing by incompressible flows.J. Eur. Math. Soc. (JEMS), 19(7):1911– 1948, 2017. (Ziyu Liu)School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, China. Email address:ziyu@ustb.edu.cn (Yankai Shi)School of Mathematical Sciences, Peking University, 100871, Beijing, China. Ema...

work page 1911

[1] [1]

Alberti, G

G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing by incompressible flows.J. Amer. Math. Soc., 32(2):445–490, 2019

work page 2019

[2] [2]

L. Arnold. Random dynamical systems. InDynamical systems (Montecatini Terme, 1994), volume 1609 of Lecture Notes in Math., pages 1–43. Springer, Berlin, 1995

work page 1994

[3] [3]

P. H. Baxendale and D. W. Stroock. Large deviations and stochastic flows of diffeomorphisms.Probab. Theory Related Fields, 80(2):169–215, 1988

work page 1988

[4] [4]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Almost-sure enhanced dissipation and uniform-in- diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes.Probab. Theory Related Fields, 179(3-4):777–834, 2021

work page 2021

[5] [5]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations.Ann. Probab., 50(1):241–303, 2022

work page 2022

[6] [6]

Bedrossian, A

J. Bedrossian, A. Blumenthal, and S. Punshon-Smith. Lagrangian chaos and scalar advection in stochastic fluid mechanics.J. Eur. Math. Soc. (JEMS), 24(6):1893–1990, 2022

work page 1990

[7] [7]

Blumenthal, M

A. Blumenthal, M. Coti Zelati, and R. S. Gvalani. Exponential mixing for random dynamical systems and an example of Pierrehumbert.Ann. Probab., 51(4):1559–1601, 2023

work page 2023

[8] [8]

A. Bressan. A lemma and a conjecture on the cost of rearrangements.Rend. Sem. Mat. Univ. Padova, 110:97–102, 2003

work page 2003

[9] [9]

B. V. Chirikov.Research concerning the theory of non-linear resonance and stochasticity. CERN, Geneva,

work page

[10] [10]

Translated at CERN from the Russian (IYAF-267-TRANS-E)

work page

[11] [11]

B. V. Chirikov. A universal instability of many-dimensional oscillator systems.Phys. Rep., 52(5):264–379, 1979

work page 1979

[12] [12]

M. Christ. Hilbert transforms along curves. I. Nilpotent groups.Ann. of Math. (2), 122(3):575–596, 1985

work page 1985

[13] [13]

Constantin, A

P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoˇ s. Diffusion and mixing in fluid flow.Ann. of Math. (2), 168(2):643–674, 2008

work page 2008

[14] [14]

Cooperman, G

W. Cooperman, G. Iyer, and S. Son. A Harris theorem for enhanced dissipation, and an example of Pierre- humbert.Nonlinearity, 38(4):Paper No. 045027, 32, 2025

work page 2025

[15] [15]

Cooperman and K

W. Cooperman and K. Rowan. Exponential scalar mixing for the 2D Navier-Stokes equations with degenerate stochastic forcing.Invent. Math., 243(3):863–959, 2026

work page 2026

[16] [16]

Coti Zelati, T

M. Coti Zelati, T. M. Elgindi, and K. Widmayer. Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow.Comm. Math. Phys., 378(2):987–1010, 2020

work page 2020

[17] [17]

Coti Zelati and V

M. Coti Zelati and V. Navarro-Fern´ andez. Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows.J. Dynam. Differential Equations, 2026, early access

work page 2026

[18] [18]

Crippa and C

G. Crippa and C. De Lellis. Estimates and regularity results for the DiPerna-Lions flow.J. Reine Angew. Math., 616:15–46, 2008

work page 2008

[19] [19]

Da Prato and J

G. Da Prato and J. Zabczyk.Ergodicity for infinite-dimensional systems. Cambridge University Press, Cam- bridge, 1996

work page 1996

[20] [20]

Dolgopyat, V

D. Dolgopyat, V. Kaloshin, and L. Koralov. Sample path properties of the stochastic flows.Ann. Probab., 32(1A):1–27, 2004

work page 2004

[21] [21]

P. Duarte. Plenty of elliptic islands for the standard family of area preserving maps.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 11(4):359–409, 1994

work page 1994

[22] [22]

T. M. Elgindi and A. Zlatoˇ s. Universal mixers in all dimensions.Adv. Math., 356:106807, 33, 2019

work page 2019

[23] [23]

Furstenberg

H. Furstenberg. Noncommuting random products.Trans. Amer. Math. Soc., 108:377–428, 1963

work page 1963

[24] [24]

T. Gallay. Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices.Arch. Ration. Mech. Anal., 230(3):939–975, 2018

work page 2018

[25] [25]

Gorodetski

A. Gorodetski. On stochastic sea of the standard map.Comm. Math. Phys., 309(1):155–192, 2012

work page 2012

[26] [26]

J. M. Greene. A method for determining a stochastic transition.Journal of Mathematical Physics, 20(6):1183– 1201, 1979

work page 1979

[27] [27]

Hairer and J

M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing.Ann. of Math. (2), 164(3):993–1032, 2006

work page 2006

[28] [28]

Hairer and J

M. Hairer and J. C. Mattingly. Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations.Ann. Probab., 36(6):2050–2091, 2008

work page 2050

[29] [29]

Hairer and J

M. Hairer and J. C. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs.Electron. J. Probab., 16:no. 23, 658–738, 2011

work page 2011

[30] [30]

Hairer and J

M. Hairer and J. C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. InSeminar on Stochastic Analysis, Random Fields and Applications VI, volume 63 ofProgr. Probab., pages 109–117. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011

[31] [31]

G. Iyer, A. Kiselev, and X. Xu. Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows.Nonlinearity, 27(5):973–985, 2014. MIXING FOR THE CHIRIKOV MAP 37

work page 2014

[32] [32]

Iyer and H

G. Iyer and H. Zhou. Quantifying the dissipation enhancement of cellular flows.SIAM J. Math. Anal., 55(6):6496–6516, 2023

work page 2023

[33] [33]

Jurdjevic.Geometric control theory, volume 52 ofCambridge Studies in Advanced Mathematics

V. Jurdjevic.Geometric control theory, volume 52 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997

work page 1997

[34] [34]

Kiselev, R

A. Kiselev, R. Shterenberg, and A. Zlatoˇ s. Relaxation enhancement by time-periodic flows.Indiana Univ. Math. J., 57(5):2137–2152, 2008

work page 2008

[35] [35]

Ledrappier

F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. InLyapunov exponents (Bre- men, 1984), volume 1186 ofLecture Notes in Math., pages 56–73. Springer, Berlin, 1986

work page 1984

[36] [36]

Lin, J.-L

Z. Lin, J.-L. Thiffeault, and C. R. Doering. Optimal stirring strategies for passive scalar mixing.J. Fluid Mech., 675:465–476, 2011

work page 2011

[37] [37]

R. S. MacKay. A renormalisation approach to invariant circles in area-preserving maps. volume 7, pages 283–300. 1983. Order in chaos (Los Alamos, N.M., 1982)

work page 1983

[38] [38]

R. S. MacKay, J. D. Meiss, and I. C. Percival. Transport in Hamiltonian systems.Phys. D, 13(1-2):55–81, 1984

work page 1984

[39] [39]

Mathew, I

G. Mathew, I. Mezi´ c, and L. Petzold. A multiscale measure for mixing.Phys. D, 211(1-2):23–46, 2005

work page 2005

[40] [40]

Meyn and R

S. Meyn and R. L. Tweedie.Markov chains and stochastic stability. Cambridge University Press, Cambridge, second edition, 2009. With a prologue by Peter W. Glynn

work page 2009

[41] [41]

Navarro-Fern´ andez and C

V. Navarro-Fern´ andez and C. Seis. Exponential mixing by random cellular flows.J. Funct. Anal., 290(2):Paper No. 111227, 2026

work page 2026

[42] [42]

C. Seis. Maximal mixing by incompressible fluid flows.Nonlinearity, 26(12):3279–3289, 2013

work page 2013

[43] [43]

S. Son. Quantitative dependence of the pierrehumbert flow’s mixing rate on the amplitude.arXiv:2410.27686, 2025

work page arXiv 2025

[44] [44]

Thiffeault

J.-L. Thiffeault. Using multiscale norms to quantify mixing and transport.Nonlinearity, 25(2):R1–R44, 2012

work page 2012

[45] [45]

D. Wei. Diffusion and mixing in fluid flow via the resolvent estimate.Sci. China Math., 64(3):507–518, 2021

work page 2021

[46] [46]

D. Wei, Z. Zhang, and W. Zhao. Linear inviscid damping and enhanced dissipation for the Kolmogorov flow. Adv. Math., 362:106963, 103, 2020

work page 2020

[47] [47]

Yao and A

Y. Yao and A. Zlatoˇ s. Mixing and un-mixing by incompressible flows.J. Eur. Math. Soc. (JEMS), 19(7):1911– 1948, 2017. (Ziyu Liu)School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, China. Email address:ziyu@ustb.edu.cn (Yankai Shi)School of Mathematical Sciences, Peking University, 100871, Beijing, China. Ema...

work page 1911